Theorem isorcl2 49619

Description: Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
isorcl.i	⊢ 𝐼 = (Iso‘𝐶)
isorcl.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
isorcl2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
isorcl2	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Proof of Theorem isorcl2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isorcl.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
2		isorcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2761	. . . . 5 ⊢ (Inv‘𝐶) = (Inv‘𝐶)
4		isorcl.i	. . . . . 6 ⊢ 𝐼 = (Iso‘𝐶)
5	4, 1	isorcl 49618	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	2, 3, 5, 4	isofval2 49617	. . . 4 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦)))
7	6	oveqd 7409	. . 3 ⊢ (𝜑 → (𝑋𝐼𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌))
8	1, 7	eleqtrd 2863	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌))
9		eqid 2761	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))
10	9	elmpocl 7633	. 2 ⊢ (𝐹 ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ dom (𝑥(Inv‘𝐶)𝑦))𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
11	8, 10	syl 17	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  dom cdm 5645  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Basecbs 17228  Invcinv 17761  Isociso 17762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-inv 17764  df-iso 17765

This theorem is referenced by:  isoval2  49620  catcisoi  49985